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1
Content available remote The Skorokhod problem with two nonlinear constraints
Li Hanwu
Probability and Mathematical Statistics
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2023
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Vol. 43, Fasc. 2
207--239
EN
In this paper, we study the Skorokhod problem with two constraints, where both constraints are nonlinear. We prove the existence and uniqueness of a solution and also provide an explicit construction for the solution. In addition, a number of properties of the solution are investigated, including continuity under uniform and J1 metrics and a comparison principle.
2
Content available remote Stochastic differential equations with constraints driven by processes with bounded p-variation
Falkowski A., Słomiński L.
Probability and Mathematical Statistics
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2015
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Vol. 35, Fasc. 2
343--365
EN
We study the existence, uniqueness and approximation of solutions of stochastic differential equations with constraints driven by processes with bounded p-variation. Our main tool are new estimates showing Lipschitz continuity of the deterministic Skorokhod problem in p-variation norm. Applications to fractional SDEs with constraints are given.
3
Content available remote Fast approximation of solutions of sde’s with oblique reflection on an orthant
Czarkowski K. T.
Probability and Mathematical Statistics
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2010
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Vol. 30, Fasc. 1
167--182
EN
We consider the discrete “fast” penalization scheme for SDE’s driven by general semimartingale on orthant Rd+ with oblique reflection.
4
Content available remote Explicit representation of the Skorokhod map with time dependent boundaries
Slaby M.
Probability and Mathematical Statistics
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2010
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Vol. 30, Fasc. 1
29--60
EN
An explicit formula for the Skorokhod type reflection map for real-valued càdlàg functions is developed in the general case of constraining set [α,β], where α and ,β are not constant but change with time. In addition, a number of properties of the reflection map, including continuity and Lipschitz conditions under uniform, J1 and M1 metrics, are studied.
5
Content available remote Discrete approximations of strong solutions of reflecting SDEs with discontinuous coefficients
Semrau A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2009
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Vol. 57, no 2
169-180
EN
We study Lp convergence for the Euler scheme for stochastic differential equations reflecting on the boundary of a general convex domain D ⊆ Rd. We assume that the equation has the pathwise uniqueness property and its coefficients are measurable and continuous almost everywhere with respect to the Lebesgue measure. In the case D = [0, ∞) new sufficient conditions ensuring pathwise uniqueness for equations with possibly discontinuous coefficients are given.
6
Content available remote Euler's approximations of weak solutions of reflecting SDEs with discontinuous coefficients
Semrau A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2007
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Vol. 55, no 2
171-182
EN
We study convergence in law for the Euler and Euler-Peano schemes for stochastic differential equations reflecting on the boundary of a general convex domain. We assume that the coefficients are measurable and continuous almost everywhere with respect to the Lebesgue measure. The proofs are based on new estimates of Krylov's type for the approximations considered.
7
Content available remote A deterministic approach to the Skorokhod problem
Bettiol P.
Control and Cybernetics
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2006
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Vol. 35, no 4
787-802
EN
We prove an existence and uniqueness result for the solutions to the Skorokhod problem on uniformly prox-regular sets through a deterministic approach. This result can be applied in order to investigate some regularity properties of the value function for differential games with reflection on the boundary.
8
Content available remote On stochastic differential equations with reflecting boundary condition in convex domains
Łaukajtys W.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2004
|
Vol. 52, nr 4
445-455
EN
Let D be an open convex set in R^d and let F be a Lipschitz operator defined on the space of adapted cadlag processes. We show that for any adapted process H and any semimartingale Z there exists a unique strong solution of the following stochastic differential equation (SDE) with reflection on the boundary of D: Xt=Ht + integral of (F(X)s-, dZs] on the interval [0, t]+Kt, t belongs to R+. Our proofs are based on new a priori estimates for solutions of the deterministic Skorokhod problem.
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